What is the easiest way to create peristalsis (when food moves inside food pipe) like animation in blender?
You could do it with Cast modifier as explained here.

But first I proposed the Lattice method:


Create your food and oesophagus objects.
Create a lattice with shift A and scale it in Object mode so that it encages a little bit more than your food.
Subdivide your Lattice in the Properties panel > Data > Lattice and U, V and W values (subdivide more than I did so that you'll be able to create details like the little pinch contraction).
Give your oesophagus a Lattice modifier with the lattice as the Object.
Select your lattice and in the Properties panel > Data > Shape keys, create a basis shapekey with the "+" button, then a second one that will be the deformation shapekey.
Keep that second shapekey selected, go in Edit mode and play with the lattice vertices so that it deforms the oesophagus the way you want.
Come back in Object mode, and play with the Shape Keys Value to deform the oesophagus.
Now, select both the lattice and the food object and move them up and down, you can see that the oesophagus deformation follows the lattice, as if the food was deforming it.
Create your animation with keyframes in the Dopesheet.